Product Description
Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups.

Topics include a description of all simply connected Lie groups in terms of semisimple Lie groups and semidirect products, the Cartan theory of complex semisimple Lie algebras, the Cartan-Weyl theory of the structure and representations of compact Lie groups and representations of complex semisimple Lie algebras, the classification of real semisimple Lie algebras, the structure theory of noncompact reductive Lie groups as it is now used in research, and integration on reductive groups. Many problems, tables, and bibliographical notes complete this comprehensive work, making the text suitable either for self-study or for courses in the second year of graduate study and beyond. ... Read more

Customer Reviews (1)

Review of Knapp's "Lie groups: beyond an introduction."
The short version:this is a superbly written and conceived book;if I had to learn this material (the basic theory of
structure and representation of Lie algebras and groups,
especially semimsimple ones) from a single book, this is
the one I'd choose, among those I've seen.If you know the
basics of abstract algebra and some very basic concepts from
topology and manifolds, and you want to learn this material,
use this book.It would be a good reference, too, as it is
easy to find things in it, and takes a fairly modern, sophisticated approach (without sacrificing motivation and
intuition).

The long version, if you want more convincing or details:

I have used several books recently in learning the structure and
representation theory of Lie algebras and groups (especially Humphreys' Introduction to Lie algebras and representation theory, Fulton
and Harris' "Representation Theory," Varadarajan's "Lie groups,
Lie algebras, and their representations.")Although I came to Knapp's book with a decent background from the others, I think it's the best pedagogically, for someone with a modicum of mathematical sophistication and some basics like abstract
algebra and an idea of what a smooth manifold is), and a smattering of Lie theory.Some examples of the book's strength:
Elementary but potentially confusing concepts (like complexification, real forms, field extensions)
are explained thoroughly but in a sophisticated way, rather
than viewed as obvious.Carefully chosen examples motivate and
clarify the general theory;consequently even though the book
is completely rigorous, and carefully delineates lemmas, proofs,
remarks, definitions, and the like, it seems less dry then some
others (e.g. Varadarajan, from my point of view).But the point
of the examples, and their relation to the general theory, is
made clear, so they do not provide an overload of detail or b
obscure the main structure.Thought is always given to the
reader's understanding, not just to logical correctness, though
the author also takes the point of view, with which I concur,
that logical clarity and sufficient detail are essential
to understanding.Relations between ideas, alternative
proofs, and the structure of the theory to come are discussed
thoroughly, but such discussion is clearly demarcated from
the main structure of the argument, so that the latter is never
obscured.This is a fantastic book, and exactly what I was
looking for.Whether you are learning the material for the
first time, or want to review it or refer to, it is a superb
source. ... Read more

Product Description

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra.

This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history.

John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994).

Customer Reviews (6)

Review of Naive Lie Theory
This review is on the textbook Naive Lie Theory by John Stillwell. Recently I purchased this book with hopes of having a study reference to the more elementary parts in preparation for more advanced study of Lie Theory and other theoretical math that involves these ideas. I have not yet finished the book. This book is well written with clear and accurate developments and good examples. There are well placed exercises. One is tempted to try various things, to explore variations based on the readings. I find this exciting the way the book let's me explore ideas. The Author lets you know about the more advanced parts of Lie Theory he is not going to cover so you have an idea what to study later to complete the picture. He decides to use simpler concepts of matrix processes and linear algebra with the understanding that this will allow you to do quite a bit. It is a nice start using the unit circle on the complex plane as an elementary first example. A clear context is given why certain inventions and discoveries were made. I am a mathematician, computer scientist, mathematical physicist, and Formal Languages.

Spectacular introduction to Lie groups and algebras
Let me start by stating my point of view: I'm a math grad student, so I'm not really the nominal audience for the book (the book is targeted toward undergraduates).Having said that, I found this book to be wonderfully conversational in tone, amusing, very honest (if there is slogging to be done in a proof, the author says so, and if the author leaves something out he tells you why), and very useful in gaining an intuitive feel for the material.The prerequisites for this book are very modest: if you've seen linear algebra and calculus, then you could give it a go.Some sort of exposure to abstract algebra of some sort would be useful, but may not be required.Some intuition for manifolds is is similarly useful, but certainly not required.

Even with these modest prerequisites, the author manages to do much with Lie Theory.This is a jewel of a book, much like its spiritual predecessor, Halmos's Naive Set Theory (Undergraduate Texts in Mathematics).

So, this book is accessible, well written and useful.What more could you ask for in an introduction?

At my level
This book is clear and neither what I already knew nor over my head.I wish the answers to the exercises were available.

a modern introduction to quantum field theory
A very good text for graduate students with a little or no knowledge of Quantum Field Theory.

Excellent read
An excellent read.In just 200 pages the author explains what Lie groups and algebras actually are.Most books on Lie theory are aimed at professional mathematicians, so begin with lots of topological and algebraic preliminaries and finally define a Lie group as a group that is also a manifold, or something similar.Stillwell begins with an example of the simplest Lie group, SO(2), as a group of rotations in the circle, then proceeds methodically to the next example SU(2), the first non-commutative Lie group.In short order all the other classical groups are discussed and, in chapter 5, the concepts of tangent space and Lie algebra are made clear through more examples.An undergraduate who has taken the calculus series, had a course in linear algebra that discusses matrices, has some knowledge of complex variables and some understanding of group theory should easily follow the material to this point.Topology, usually a graduate topic, is introduced later while showing which Lie groups are simply-connected, and how this is used to distinguish between similar Lie groups.

The material was clearly discussed and I found only a couple of typos.But I also found the use of the word vector and matrix for the same object in the same paragraph somewhat dis-quieting.Lastly, I would have liked to have seen some mention of Lie theory connections with modern physics.

... Read more

Product Description

This book provides the reader with the tools to understand the ongoing classification and construction project of Lie superalgebras. It presents the material in as simple terms as possible. Coverage specifically details Borcherds-Kac-Moody superalgebras. The book examines the link between the above class of Lie superalgebras and automorphic form and explains their construction from lattice vertex algebras. It also includes all necessary background information.

Product Description
This is the softcover reprint of the English translation of 1975 (available from Springer since 1989) of the first 3 chapters of Bourbaki's 'Groupes et algèbres de Lie'. The first chapter describes the theory of Lie algebras, their derivations, their representations and their enveloping algebras. In Ch. 2, free Lie algebras are introduced in order to discuss the exponential, logarithmic and the Hausdorff series. Ch. 3 deals with the theory of Lie groups over R and C and ultrametric fields. It describes the connections between their local and global properties, and the properties of their Lie algebras. It is one of the very best references on this subject. ... Read more

Customer Reviews (1)

great as references, and..
...even better when supplemented by other texts (be wary though, as the notation of Bourbaki is not universally accepted).
I'd buy themsimply for their sparkling clarity and careful attention to rigour (esp. Commalg, Lie theory and integration).
These arewonderfully crafted, masterful expositions, and shouldnot, by any means, be overlooked by the (aspiring) pure mathematician.

Cheers,
A. ... Read more

Product Description
This is an introduction to Lie algebras and their applications in physics. First illustrating how Lie algebras arise naturally from symmetries of physical systems, the book then gives a detailed introduction to Lie algebras and their representations, covering the Cartan-Weyl basis, simple and affine Lie algebras, real forms and Lie groups, the Weyl group, automorphisms, loop algebras and highest weight representations. The book also discusses specific further topics, such as Verma modules, Casimirs, tensor products and Clebsch-Gordan coefficients, invariant tensors, subalgebras and branching rules, Young tableaux, spinors, Clifford algebras and supersymmetry, representations on function spaces, and Hopf algebras and representation rings. A detailed reference list is provided, and many exercises and examples throughout the book illustrate the use of Lie algebras in real physical problems. The text is written at a level accessible to graduate students, but will also provide a comprehensive reference for researchers. ... Read more

Customer Reviews (1)

Mixed feelings
Lie groups and Lie algebras permeate most parts of theoretical physics. Every student in physics should have some basic notions of the subject as it sometimes tends to have unsuspected applications.

The first three chapters of this book include exemples and motivation for the more formal aspect of the Lie theory. Those are also meant to set the notation used later throughout the book. Topics covered should be well-known from a senior undergraduate student with a good background in quantum mechanics (harmonic oscillator, the rotation group) and particle physics (mostly the "zoological" part of it : classification of particles, the eightfold way and so on).
From chapter 4 on, the Maths definitely take the most prominent part of the stage. Chapter 4 is a reminder of basic notions in algebra, as covered in an undergraduate course in algebra and classical groups.
Chapter 5, on representation, should not be a challenge to the physicist.
The core of the subject is presented in chapter 6, where the idea of the Cartan-Weyl basis is given a nice presentation. This chapter is a little bit more demanding. Some statements are not proved. However, a committed student in physics, should be able to devise proofs for him/herself.
Chapter 7 is particularly enjoyable, dealing with Dynkin diagrams and the classification of finite simple Lie algebras, and introducing infinite dimensional ones. The way Kac-Moody algebras appear, through relaxing the axioms of the Chevalley-Serre construction should be appreciated. Also, physical exemples are to the point.
However, beginning with chapter 12, the wrongs of this book become somewhat annoying. For instance, in chapter 12, the authors of this book freely speak of Verma modules, highest weight representations, while these concepts are to be introduced and properly developped in later chapters. I found this chaffing from an introductory book. From chapter 12, it seems that the reader is to gently follow and accept the statements made by the author, without encountering much proof or hint to this all.
Things come more acceptable in later chapters only, where invariant tensors and other things more familiar from a physicist with no previous acquaintance to Lie algebras, are exposed.

All in all, a good book for some parts of it but whose value could have surely been enhanced by adopting a more pedagogical presentations. Some proofs to key facts in the more "exotic subjects", would have been welcome, too. All the more, that some chapters of this book did not require much work from the authors, as it seems that they were taken from Dr. Fuchs "Affine Lie algebras".
Hopefully, welcome additions will be added to a further edition.
Beginners or readers with a casual interest in Lie algebras should better learn it from another source.
... Read more

Product Description
This volume presents lecture notes based on the author's courses on Lie algebras and the solution of Hilbert's fifth problem. In chapter 1, "Lie Algebras," the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, "The Structure of Locally Compact Groups," deals with the solution of Hilbert's fifth problem given by Gleason, Montgomery, and Zipplin in 1952. ... Read more

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Semisimple Lie groups, and their algebraic analogues over fields other than the reals, are of fundamental importance in geometry, number theory, and mathematical physics. Three independent, self-contained volumes, under the general title 'Lie Theory,' feature survey work and original results by well-established researchers in key areas of semisimple Lie theory. A wide spectrum of topics is treated, with emphasis on the interplay between representation theory and the geometry of adjoint orbits for Lie algebras over fields of possibly finite characteristic, as well as for infinite-dimensional Lie algebras. Also covered is unitary representation theory and branching laws for reductive subgroups, an active part of modern representation theory. Finally, there is a thorough discussion of compactifications of symmetric spaces, number theory via Selberg's trace formula, and harmonic analysis through a far-reaching generalization of Harish-Chandra's Plancherel formula for semisimple Lie groups. Ideal for graduate students and researchers, 'Lie Theory' provides a broad, clearly focused examination of semisimple Lie groups and their integral importance to research in many branches of mathematics. 'Lie Theory: Lie Algebras and Representations' contains J. C. Jantzen's Nilpotent Orbits in Representation Theory,' and K.-H. Neeb's 'Infinite Dimensional Groups and their Representations.' Both are comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. 'Lie Theory: Unitary Representations, Number Theory, and Compactifications' contains work by A. Borel and L. Ji, T. Kobayashi and J.-P. Labesse. 'Lie Theory: Harmonic Analysis on Symmetric Spaces' features work by E. van den Ban, P. Delorme, and H. Schlichtkrull. ... Read more

Product Description
The representation theory of affine lie algebras has been developed in close connection with various areas of mathematics and mathematical physics in the last two decades. There are three valuable works on it, written by Victor G Kac. This volume begins with a survey and review of the material treated in Kac's books. In particular, modular invariance and conformal invariance are explained in more detail. The book then goes further, dealing with some of the recent topics involving the representation theory of affine lie algebras. Since these topics are important not only in themselves but also in their application to some areas of mathematics and mathematical physics, the book expounds them with examples and detailed calculations. ... Read more

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This is the softcover reprint of the English translation of Bourbaki's text Groupes et Algèbres de Lie, Chapters 7 to 9. It completes the previously published translations of Chapters 1 to 3 (3-540-64242-0) and 4 to 6 (978-3-540-69171-6) by covering the structure and representation theory of semi-simple Lie algebras and compact Lie groups. Chapter 7 deals with Cartan subalgebras of Lie algebras, regular elements and conjugacy theorems. Chapter 8 begins with the structure of split semi-simple Lie algebras and their root systems. It goes on to describe the finite-dimensional modules for such algebras, including the character formula of Hermann Weyl. It concludes with the theory of Chevalley orders. Chapter 9 is devoted to the theory of compact Lie groups, beginning with a discussion of their maximal tori, root systems and Weyl groups. It goes on to describe the representation theory of compact Lie groups, including the application of integration to establish Weyl's formula in this context. The chapter concludes with a discussion of the actions of compact Lie groups on manifolds. The nine chapters together form the most comprehensive text available on the theory of Lie groups and Lie algebras.

Product Description
The main goal of this book is to present an introduction to and applications of the theory of Hopf algebras. The authors also discuss some important aspects of the theory of Lie algebras.The first chapter can be viewed as a primer on Lie algebras, with the main goal to explain and prove the Gabriel-Bernstein-Gelfand-Ponomarev theorem on the correspondence between the representations of Lie algebras and quivers; this material has not previously appeared in book form.The next two chapters are also "primers" on coalgebras and Hopf algebras, respectively; they aim specifically to give sufficient background on these topics for use in the main part of the book. Chapters 4-7 are devoted to four of the most beautiful Hopf algebras currently known: the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups (although these two are isomorphic, they are very different in the aspects they bring to the forefront), the Hopf algebras of the nonsymmetric and quasisymmetric functions (these two are dual and both generalize the previous two), and the Hopf algebra of permutations. The last chapter is a survey of applications of Hopf algebras in many varied parts of mathematics and physics.Unique features of the book include a new way to introduce Hopf algebras and coalgebras, an extensive discussion of the many universal properties of the functor of the Witt vectors, a thorough discussion of duality aspects of all the Hopf algebras mentioned, emphasis on the combinatorial aspects of Hopf algebras, and a survey of applications already mentioned. The book also contains an extensive (more than 700 entries) bibliography. ... Read more

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Devoted to an important and popular branch of modern theoretical and mathematical physics, this book introduces the use of Lie algebra and differential geometry methods to study nonlinear integrable systems of Toda type. Many challenging problems in theoretical physics are related to the solution of nonlinear systems of partial differential equations. One of the most fruitful approaches in recent years has resulted from a merging of group algebraic and geometric techniques. The book gives a comprehensive introduction to this exciting branch of science. Chapters 1 and 2 review basic notions of Lie algebras and differential geometry with an emphasis on further applications tointegrable nonlinear systems. Chapter 3 contains a derivation of Toda type systems and their general solutions based on Lie algebra and differential geometry methods. The last chapter examines explicit solutions of the corresponding equations. The book is written in an accessible "lecture note" style with many examples and exercises to illustrate key points and to reinforce understanding. ... Read more

Product Description
This volume is devoted to the theory of nilpotent Lie algebrasand their applications. Nilpotent Lie algebras have played animportant role over the last years both in the domain of algebra,considering its role in the classification problems of Lie algebras,and in the domain of differential geometry. Among the topics discussedhere are the following: cohomology theory of Lie algebras,deformations and contractions, the algebraic variety of the laws ofLie algebras, the variety of nilpotent laws, and characteristicallynilpotent Lie algebras in nilmanifolds.
Audience: This book is intended for graduate studentsspecialising in algebra, differential geometry and in theoreticalphysics and for researchers in mathematics and in theoretical physics. ... Read more

Customer Reviews (1)

Interesting title, quite bad book.
The title of this book does not properly correspond to its contents. It is supposed to develop the general theory of nilpotent Lie algebras, but the most part of the book deals with a quite special class of nilpotent algebras, called filiform. Only the first two chapters present some general material, along with a list of nilpotent algebras in low dimensions which is full of mistakes. On the other hand, the authors use many different notations for the same objects, which confuses the reader. The quantity of wrongly ennounced results (due to misprints, etc) is discouraging, and shows how careless the final text has been prepared by the editors. This suffices to give it up.
It is a pity, since there is no book dealing with nilpotent algebras only, but certainly this is not the way to write it. ... Read more

Product Description
This self-contained text concentrates on the perspective of analysis, assuming only elementary knowledge of linear algebra and basic differential calculus. The author describes, in detail, many interesting examples, including formulas which have not previously appeared in book form. Topics covered include the Haar measure and invariant integration, spherical harmonics, Fourier analysis and the heat equation, Poisson kernel, the Laplace equation and harmonic functions. Perfect for advanced undergraduates and graduates in geometric analysis, harmonic analysis and representation theory, the tools developed will also be useful for specialists in stochastic calculation and the statisticians. With numerous exercises and worked examples, the text is ideal for a graduate course on analysis on Lie groups. ... Read more

Product Description
Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The Appendix provides a summary of the basic properties of each Lie algebra of finite and affine type. ... Read more

Product Description
This is an introductory text on Lie groups and algebras and their roles in diverse areas of pure and applied mathematics and physics. The material is presented in a way that is at once intuitive, geometric, applications oriented, and, most of the time, mathematically rigorous. It is intended for students and researchers without an extensive background in physics, algebra, or geometry. In addition to an exposition of the fundamental machinery of the subject, there are many concrete examples that illustrate the role of Lie groups and algebras in various fields of mathematics and physics: elementary particle physics, Riemannian geometry, symmetries of differential equations, completely integrable systems, and bifurcation theory. ... Read more

Customer Reviews (1)

An interesting introduction
This book is intended as a first introduction to the theory of Lie groups and Lie algebras, focused on applications in physics. In its first chapters the authors introduce the material basing on important examples like the rotation algebra or the realization of the Heisenberg Lie algbebra in terms of annihilation/creation operators. This will lead to the general theory, having in mind these important physical examples. The book is essentially divided into three parts. The first is a differential geometric chapter dealing with the usual concepts of manifolds, vector fields, integration theorems, etc, but it also provides a topic which is usually not covered by textsbooks, namely the Maurer-Cartan equations of a Lie group/algebra. This is an important alternative, both from the physical as from the mathematical point of view. The second part corresponds to the algebraic theory of complex semisimple Lie algebras, which corresponds more or less to the standard contents of a textbook. The authors also briefly comment on the real forms of complex simple Lie algebras, which is an essential ingredient for physical applications (see e.g. the kinematical Lie algebras). The third part corresponds to representation theory of complex semisimple algebras, motivated by a detailed exposition of the eightfold way of Gell-Mann and Ne'eman. The study of representations also analyzes briefly tensor product decompositions. A final chapter is devoted to the application of Lie groups to the integration of Hamiltonian systems, a topic which has become of great interest in the last years. This kind of appendix is a good introduction to more advanced expositions like the monography of Fomenko and Trofimov.
Globally, the book covers the most important topics on Lie algebras/groups that are necessary for physical applications. Many proper notations like Pauli and Gell-Mann matrices are used, and each section is completed with a set of exercises. The book presents only very few misprints, like in the tensor product of the standard representation of the su(3) algebra.
It is very recommendable as an introductory text to Lie theory. ... Read more

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Lie algebras are efficient tools for analyzing the properties of physical systems. Concrete applications comprise the formulation of symmetries of Hamiltonian systems, the description of atomic, molecular and nuclear spectra, the physics of elementary particles and many others. This work gives an introduction to the properties and the structure of the Lie algebras su(n). First, characteristic quantities such as structure constants, the Killing form and functions of Lie algebras are introduced. The properties of the algebras su(2), su(3) and su(4) are investigated in detail. Geometric models of the representations are developed. A lot of care is taken over the use of the term 'multiplet of an algebra'.The book features an elementary (matrix) access to su(N)-algebras, and gives a first insight into Lie algebras. Student readers should be enabled to begin studies on physical su(N)-applications, instructors will profit from the detailed calculations and examples. ... Read more

Customer Reviews (4)

Really about time su(n) representaions like these were available
For me the su(2),su(3) material wasn't new.For me the su(2),su(3) material wasn't new.
I was grateful for the representation of su(4) and the structure constants.
The expansion of the Gell-Mann matrices is consistent with standard physic text
notation ( so the su(4) structure constants in Gordon Kane's "Modern Elementary Particle Physics" agree exactly).
The explanation of Young tableau is welcome andthe subgroup structures given
illuminate the Dynkin diagrams for these Lie algebras as well.
I find myself wishing this text were available in the 70's or 80's.
The exposition is clear and covers the material well.
One needs some modern algebra background and a familiarity with matrix representation notation.His explanation of how two so(n) groups are Hermitian in su(n) is something it took me years to figure out on my own!
I say this is a book well done and one that has been needed for a long time.
I would that he had expanded the book with an su(5) representations, structure constants
and sub-algebras
I was grateful for the representaion of su(4) and the structure constants.
The expansion of the Gell-Mann matrices is consistent with standard physic text notation ( so the su(4) structure constants in Gordon Kane's "Modern Elementry Particle Physics" agree exactly).
The explaination of Young tableau is welcome andthe subgroup structures givenilluninate the Dynkin diagrams for these Lie algebras as well.
I find myself wishing this text were available in the 70's or 80's.
The exposition is clear and covers the material well.
One needs some modern algebra backgrond and a familarity with matrix representaion notation.
His explaination of how two so(n) groups are Hermetian in su(n) is something it took me years to figure out on my own!
I say this is a book well done and one that has been needed for a long time.I would that he had expanded the book with an su(5) representations, structure constants and sub-algebras

GREAT!
Great indtroductory book, very user friendly.It explains and shows in an easy to understand and simple way....lots of examples and explanations. EASY AND ENJOYABLE READ.

Lie algebra demystified
A practical introduction to an esoteric topic which frightens many physics students.

This book presupposes little background mathematics and begins by defining lie alegebras and providing adequate examples. He then details some basic properties of finite dimensional lie algebras and offers several ways of "representing" them including the adjoint representation. From the beginning there is an emphasis on applications to quantum mechanics and I especially enjoyed the section on SU ( 2) and it's application to angular momentum operators. SU ( 3) and SU ( 4 ) are developed in due time in a logical and easy to understand format.

He also shows, in a simple way, how the tangent space of the identity of a lie group has a lie algebra structure which is useful in studying the group's local properties.

A very handy reference for those studying advanced quantum mechanics and particle physics yet basic enough for undergraduates to grasp the concepts.

An excellent practical guide
This short book covers an important aspect that has been neglected by most textbooks on Lie algebras written for physicists, namely providing a comprehensible introduction for undergraduates based on detailed examples, computations and precise motivations, without having to develop the formal theory. This is not a textbook on Lie algebras in the usual sense, but a practical guide whose intention is to provide a solid comprehension of the main facts on (finite dimensional) Lie algebras used in physics. This justifies the choice of the objects analyzed, the compact real form su(N) of the Lie algebras sl(N,C), which constitute an essential tool in the study of the interacting boson model and nuclear rotational states. The topics covered by this book are quite modest (there are no general proofs and no development of classical problems like the classification of simple Lie algebras), and focuses on a detailed comment on the properties of simple algebras using mainly three Lie algebras, su(2),su(3) and su(4), before ennouncing the general case in the last chapter. However, this should not be understated, specially because the book explains carefully the usual notations (which change in the literature from author to author) and tries to clarify the reasons that justify the study of the formal theory.
The book is divided into six chapters, which we comment separately. The first chapter is a quick and effective overview on the basic properties of simple Lie algebras, namely the adjoint representation, the Killing form, representations and their reducibility. For the inner product the Dirac bracket notation is used. The concept of multiplets, which plays an essential role, is introduced at the end of this chapter. Chapter 2 begins with a short discussion of hermitian matrices, and introduces the Lie algebra su(N) in the usual way. The complexification of this algebra is shortly commented, as well as the generation of the algebra by means of operators. The structure constants over the standard basis are obtained, and as application the Killing form for su(N) is computed. It should be said that the notations used in this chapter have in mind the Gell-Mann matrices, which will be introduced later. Chapter 3 studies the fundamental facts concerning the rank one algebra su(2), and which will be central to later developments. The topics commented are generators of su(2), that is, the Pauli matrices, the quantum mechanical operators J of angular momentum, the su(2) multiplets and the irreducible (complex) representations. Further the tensor products (called "direct products") of these representations and their decomposition into irreducible components is commented. Many very detailed computations are presented, which illustrate clearly the procedure and its significance. Moreover the graphical method for the tensor product decomposition is developed,
The fourth chapter, devoted to the Lie algebra su(3), which cosntitutes in some sense the core of this book, actually develops the main aspects necessary to the description of global symmetry schemes for hadrons (without deeping into the actual classification, for this would require a basic knowledge of quantum field theory). The Lie algebra su(3) is introduced according Gell-Mann's notation. The step operators and states of su(3) are introduced, and the individual states and multiplicities are carefully constructed using graphical motivation (which actually corresponds to the standard application of the su(2)-triples). In order to formalize the construction, the Young tableaux are used (these constituting an essential tool for the analysis of the su(N) algebras). Special attention is devoted to the fundamental su(3)-multiplet (the quark representation 3) and its dual. This leads naturally to the introduction of the hypercharge Y (however no reference to the Gell-Mann-Nishijima formula is made). The (quadratic) Casimir operator of su(3) and its eigenvalues are analyzed, with explicit examples that point out the main properties of this invariant. The next section focuses on the tensor products of su(3)-multiplets, and develops also the graphical method to deduce the decomposition. A table presents some of these tensor products (for highest weights lower or equal to (2,1)). Again, this motivation is used to present the Young tableaux. Chapter 5 presents more or less the same topics for the rank three algebra su(4), and discusses the charm C (as a natural consequence of the quantum numbers discussed for su(3)). The multiplets and tensor products are reviewed (the diagrams are of exceptional quality and clarity), and the chapter finishes commenting on the standard Weyl basis (that is, the basis obtained from the root system of the corresponding algebra; this is the presentation that will be found in almost any book on Lie algebras). These facts are presented without proof, but serve to illustrate fundamental facts like the Cartan integers or the presentation by generators and relations that the interested reader will find in any standard text. Chapter six gives a recopilation of the basic facts of the su(N) algebras for arbitrary values of N (hermitian generators and multiplets, quadratic Casimir operator, etc). The bibliography presents some texts to profound the study. A little remark: the reference to Cornwell's book refers specifically to volume II, which deals with the theory of finite dimensional Lie algebras.
On balance I think this book is an excellent first contact with Lie algebras for those using them in physics, because of the lucid style and the clarity in the exposition. The very detailed calculations and step by step introduction of the material allow the readers not familiar with Lie algebras tobecome confident with the main facts they will find in any standard textbook, and which often discourages because of notational problems or implicit assumption of knowledge concerning the fundamental properties. Although the notation is mainly that used in physics literature, the examples and motivations introduced in this text will help the reader in the transition to other books using alternative notations. This work is a welcome reference for both beginners in Lie algebras for physics, as well as for instructors. ... Read more